The scaling theory of [1]} was extended to 3D low-Reynolds number channel flows with finite longitudinal gratings by [2]}.
They used a long-wave limit and assumed low surfactant concentrations to show that the slip velocity and the slip length scale in a similar fashion as in 2D flows.
In the limit of large Damköhler numbers, as found for common surfactants, and sufficiently small bulk Péclet numbers, they predicted that significant slip can be achieved provided that the grating length is longer than both a modified depletion length and a mobilization length.
The modified depletion length depends on surfactant properties and the height of the channel, and is generally small, of the order of 1 mm, for small-scale applications.
The mobilization length depends on the normalised surfactant concentration, Marangoni number, Damköhler number and Biot number.
Since the mobilization length can be much longer for typical small-scale applications, they concluded that the mobilization length alone can control the slip in most small-scale applications.
If the grating length is smaller than the mobilization length, they showed that the slip velocity increases with the square of the grating length, which is in agreement with their numerical simulations and experiments, as well as with the experimental results from [3]} and [4]}.
